Abstract
We characterize Lp norms of functions onR n for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of Lp functions converge to the functions almost everywhere and in Lp for 1<p<∞. In L1 we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L1 in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for Lp (R n) when 1≤p<∞.
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Communicated by John J. Benedetto
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Grafakos, L., Lennard, C. Characterization ofL p (R n) using Gabor frames. The Journal of Fourier Analysis and Applications 7, 101–126 (2001). https://doi.org/10.1007/BF02510419
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DOI: https://doi.org/10.1007/BF02510419